PROPER AFFINE HYPERSPHERES WHICH FIBER ...

c 2003 International Press !

ASIAN J. MATH. Vol. 7, No. 1, pp. 115–132, March 2003

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PROPER AFFINE HYPERSPHERES WHICH FIBER OVER ¨ PROJECTIVE SPECIAL KAHLER MANIFOLDS ∗ ´ ‡ OLIVER BAUES† AND VICENTE CORTES Abstract. We show that the natural S 1 -bundle over a projective special K¨ ahler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration S2n+1 −→ CPn in the context of projective special K¨ ahler manifolds. As an application we have that a natural circle bundle over the Kuranishi moduli space of a Calabi-Yau threefold is a Lorentzian proper affine hypersphere.

Introduction. In a previous paper [BC], we proved that any simply connected special K¨ ahler manifold admits a canonical immersion into affine space as a parabolic affine hypersphere. A particular important class of special K¨ ahler manifolds are conic special K¨ ahler manifolds. These are by definition special K¨ ahler manifolds which are locally modelled on a complex cone over some complex projective manifold which is then called a projective special K¨ ahler manifold. The purpose of this paper is to provide an understanding of the particular (affine) differential geometry which is canonically associated with projective special K¨ ahler manifolds. Whereas the conic special K¨ ahler manifold M which is associated with a simply ¯ carries the geometry of a parabolic connected projective special K¨ ahler manifold M (or improper) affine hypersphere, we show that the total space S of a natural circle ¯ is a proper affine hypersphere. The S 1 -action on S induces a Sasakian bundle S → M structure on S which is compatible with the affine differential geometry in a very specific sense. Moreover, all information about the conic special K¨ ahler geometry on M is encoded in the affine Sasakian geometry on S. Lu showed [L] that every complete affine special K¨ ahler manifold with a positive definite metric is flat. Using a well known result of Calabi [Ca2] on complete convex affine hyperspheres we obtain an analogous result for projective special K¨ ahler ¯ is a (simply connected) complete projective special manifolds: We show that if M ¯ is isometric to CPn with K¨ ahler manifold with a definite affine metric on S then M the canonical Fubini-Study metric. The construction of the affine sphere S over a projective special K¨ ahler manifold naturally relates to well known canonical data on the Kuranishi moduli space for Calabi-Yau three-manifolds. Thereby we show that a natural circle bundle over the Kuranishi moduli space admits a canonical structure of an affine hypersphere with affine metric of Lorentzian signature. ¯ is complete, and the metric on S is not definite, as in the case of Kuranishi If M moduli spaces, then interesting complete models for projective special K¨ ahler mani¯ which admit a transitive semisimple folds do exist. We describe all fibrations S → M group of automorphisms preserving the projective special K¨ ahler structure on the base ¯ . These are particular examples of homogeneous Lorentzian affine hyperspheres M fibering over Hermitian symmetric spaces.

∗ Received

November 28, 2002; accepted for publication February 10, 2003. Mathematik, ETH-Zentrum, R¨ amistrasse 101, CH-8092 Z¨ urich, Switzerland ([email protected]). ‡ Institut Elie ´ Cartan, Universit´ e Henri Poincar´ e-Nancy I, B.P. 239, F-54506 Vandœuvre-l` esNancy, France ([email protected]). † Departement

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´ O. BAUES AND V. CORTES

116 1. Preliminaries.

1.1. Affine hypersurfaces. For the convenience of the reader, and to fix the notation, we recall the basic definitions of affine differential geometry of hypersurfaces in Rn+1 and the definition of affine hyperspheres. For more details, see for example [NS, Ca2]. Let det denote the standard volume form on Rn+1 , and ∇ the standard flat connection on Rn+1 . In the context of affine immersions we consider manifolds ˆ so that with a semi-Riemannian metric g and a torsionfree connection ∇ ˆ i) the cubic tensor ∇g is totally symmetric, and ˆ ii) the metric volume form θg is ∇-parallel. ˆ are then said to satisfy the compatibility condition i) and the equiaffine The data (g, ∇) condition ii). Every nondegenerate hypersurface immersion ψ : M → Rn+1 induces ˆ g) on M which satisfy i) and ii) via the fundamental formula data (∇, ˆ X Y + g(X, Y )E , ∇X Y = ∇

(1)

where X, Y denote vector fields on M , and E is the affine normal of the immersion. (Note that the notation identifies M as a submanifold of Rn+1 .) The affine normal E is a canonical normal vector field along ψ which is defined up to sign by the condition ˆ g) satisfies ii), and the normalizing condition that the pair (∇, iii) det(E, . . .) = θg on M . ˆ g) → The metric g is then called the Blaschke metric and the immersion ψ : (M, ∇, Rn+1 a Blaschke immersion. The tensor A = −∇E is horizontal along ψ and is called the shape tensor of the immersion. The quantity H = n1 trA is called the affine mean ˆ is flat and n > 1 then, by the equation of Gauß, A = 0 and the affine curvature. If ∇ normal is the restriction of a constant vector field. In this case, ψ is called a parabolic (or improper) affine hypersphere. If the shape tensor equals a constant multiple of the identity, A = κ id, where κ $= 0, ψ is called a proper affine hypersphere. In this case, ˆ is projectively flat. An affine hypersphere has constant mean curvature H = κ. ∇ ˆ g) which satisfy i) and ii). We put ∇ ˆ ∗ for the Let M be a manifold with data (∇, ˆ conjugate connection of ∇ with respect to g. It is torsionfree by the compatibility condition i). Then the fundamental theorem of affine differential geometry asserts that ˆ g) arises from a Blaschke immersion ψ a simply connected manifold M with data (∇, if and only if the integrability condition ˆ ∗ is projectively flat iv) ∇ ˆ g) up to composition is satisfied. The immersion ψ is determined by the data (∇, ˆ is flat. Then with an unimodular affine transformation. A special case arises if ∇ ∗ ˆ it is easily seen that ∇ is also flat. Hence, iv) is satisfied and M is a parabolic ˆ g) arise from an immersion affine hypersphere. We also mention that the data (∇, ˆ has totally symmetric as an affine sphere if and only if the cubic tensor C = ∇g ˆ ˆ derivative ∇C. If (M, ∇, g) is a manifold which satisfies the integrability conditions for a Blaschke immersion as an affine sphere we say that M has the structure of an affine sphere. 1.2. Special K¨ ahler manifolds. We recall some basic notions and constructions from special K¨ ahler geometry. For more details the reader can consult [ACD], and also [F]. A special K¨ ahler manifold (M, J, g, ∇) is a (pseudo-) K¨ ahler manifold (M, J, g) together with a flat torsionfree connection ∇ such that ∇ω = 0, where ω = g(·, J·) is the K¨ ahler form, and such that ∇J is symmetric, i.e. d∇ J(X, Y ) := (∇X J)Y − (∇Y J)X = 0 for all vector fields X and Y .

¨ AFFINE HYPERSPHERES AND SPECIAL KAHLER MANIFOLDS

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More precisely, one should speak of affine special K¨ ahler manifolds since there is also the notion of a projective special K¨ ahler manifold. In fact, there is a class of (affine) special K¨ ahler manifolds (M, J, g, ∇), which are called conic special K¨ ahler manifolds and which are characterized by the existence of a local holomorphic C∗ -action ϕλ : M → M , λ = reit ∈ C∗ , with the property: (ϕλ )∗ X = r cos tX + r sin tJX for all ∇-parallel vector fields X on M . Under appropriate regularity assumptions on the action, the projection ¯ = P (M ) π : M −→ M ¯ = P (M ) is a holomorphic submersion onto a complex onto the space of orbits M ¯ inherits a (pseudo-) K¨ (Hausdorff-) manifold. Then M ahler metric g¯ from (M, g), ¯ and the base (M , g¯) is called a projective special K¨ ahler manifold. Although, strictly speaking, the fully fledged projective special K¨ ahler geometry is encoded in the geo¯. metric data on the bundle π : M → M Special K¨ ahler manifolds may also be characterized in terms of complex Lagrangian immersions (see [ACD]). In fact, any simply connected special K¨ ahler manifold (M, J, g, ∇) has a canonical realization as a (pseudo-) K¨ ahlerian immersed Lagrangian submanifold of a pseudo-Hermitian, complex symplectic vector space (V, γ, Ω) with split signature. This means that there exists a holomorphic Lagrangian immersion λ : M → V so that g = λ∗ γ is the pull-back of the hermitian product γ. Moreover, the projection onto the subspace V τ of real points for the real structure τ defined by the relation Ω = −i γ(·, τ ·) gives local flat coordinates on M which determine the flat connection ∇. The holomorphic Lagrangian immersion λ is determined by the data (g, ∇) up to a complex affine transformation which preserves γ and Ω. Conic special K¨ ahler manifolds may be realized by immersions λ which are equivariant with respect to the natural C∗ -action on V . λ is then uniquely determined up to a complex linear transformation which preserves γ and Ω. We then call λ a compatible Lagrange immersion of the (conic) special K¨ ahler manifold M . 2. The local geometry. It is well known that holomorphic Lagrangian immersions λ into a complex 2n-dimensional symplectic vector space V are locally of the form λ = λF := dF : U → T ∗ Cn ∼ = V , where F is a holomorphic function defined on some domain U ⊂ Cn . The K¨ ahler condition for the holomorphic Lagrangian immersion λF is an open condition on the real 2-jet of F . Conic special K¨ ahler manifolds correspond to potentials which are homogeneous of degree 2. Therefore the local geometry of (conic) special K¨ ahler manifolds may be described in terms of a holomorphic potential F . Special K¨ ahler domains. Let U ⊂ Cn be a connected open domain and F : U → C a holomorphic function which satisfies the condition that the matrix ! 2 " ∂ F Im ∂zi ∂zj is nondegenerate. Then the function !# " ∂F 1 z¯i k = Im 2 ∂zi

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¯ and defines a K¨ ahler potential on U . With the corresponding K¨ ahler form ω = i ∂ ∂k, 1 metric g = ω(i ·, ·), the domain U is a (pseudo-) K¨ ahler manifold . Such a domain U will be called a special K¨ ahler domain. On a special K¨ ahler domain U there are flat coordinates, called flat special coordinates, xi = Re(zi ) , yi = Re(

∂F ) ∂zi

(2)

which define on U a torsionfree flat connection ∇ so that ω is parallel. The complex manifold U with the data (g, ∇) is then a special K¨ ahler manifold. Conversely, any special K¨ ahler manifold is locally equivalent to a special K¨ ahler domain (U, g, ∇). Another peculiar feature of special K¨ ahler domains is that the K¨ ahler metric g is a Hessian metric with respect to the flat connection. This means that on U there exists a real potential function f so that g = ∇df . (The fact that g is locally Hessian is well known. An explicit formula for f which is given in terms of the holomorphic function F , see [C2], shows that f exists globally on U .) Moreover, in the flat coordinates the smooth function f satisfies the Monge-Amp`ere equation | det ∂ 2 f | = c ,

(3)

where c > 0 is a constant. As a consequence, the data (g, ∇) give U the geometry of a parabolic affine hypersphere, see [BC]. Explicitly, λ(u) = (x1 (u), . . . , xn (u), y1 (u), . . . , yn (u), f (u)) defines a Blaschke immersion λ : U → R2n+1 into affine space R2n+1 which induces the data (∇, g). 2.1. The metric geometry of conic special K¨ ahler domains. In this paper, we are mainly concerned with conic special K¨ ahler domains. We call a special K¨ ahler domain U ⊂ Cn+1 \{0} conic, if C∗ U ⊂ U and if the holomorphic prepotential F is a homogeneous function of degree 2. Moreover, we require that the potential k does not vanish on a conic special K¨ ahler domain. Locally, any conic special K¨ ahler manifold is equivalent to a conic special K¨ ahler domain U ⊂ Cn+1 \{0}. To any conic domain ¯ denote its image in the projective space CPn . We consider the U ⊂ Cn+1 we let U projection map ¯ π : U −→ U ¯ . The special which is a submersion, and view U as a principal C∗ -bundle over U ¯ K¨ ahler metric g on U naturally induces a K¨ ahler metric g¯ on U via the projection π. The metric g¯ is defined by the formula $ $2 gu (X, X) $$ gu (X, u) $$ −$ , X ∈ Tu Cn+1 . (4) g¯π(u) (dπ(X), dπ(X)) = gu (u, u) gu (u, u) $

(Note that g is definite on the vertical spaces Vu = Cu ⊂ Tu Cn+1 of the fibration π by the condition that k $= 0, see Lemma 2 below.) Let ω ¯ denote the corresponding ¯ . Then it is easy to see that the pull-back π ∗ ω K¨ ahler form on U ¯ on U is given by π∗ ω ¯ = i ∂ ∂¯ log k 1 We

do not require that the K¨ ahler metric g is definite


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¯ ⊂ CPn with the metric g¯ a on the horizontal space Hu = Vu⊥ . We call the domain U projective special K¨ ahler domain. The simplest example of such a domain is projective space CPn itself with the Fubini-Study metric: % Example 2.1. Putting U = Cn+1 \{0} and F (z0 , . . . , zn ) = i zj2 , formula (4), ¯ = CPn . The famous Hopf-fibration defines the Fubini-Study metric on U S2n+1 −→ CPn exhibits the sphere S2n+1 = {u ∈ Cn+1 | |u|2 = 1} as a S1 -principal bundle over CPn . The Hopf fibration is also known to be a Riemannian submersion with respect to the standard metric on the sphere if the metric on CPn is suitably normalized. It is the content of our next proposition that the geometric construction of the Hopf-fibration generalizes in the context of projective special K¨ ahler domains. To establish this result we consider now the K¨ ahler potential k on U . We remark that k satisfies k(αu) = |α|2 k(u), for α ∈ C∗ , and, by assumption, never vanishes on U . We put Mc = {u ∈ U | |k(u)| = c}. Then the level surface Mc is a real hypersurface in U ⊂ Cn+1 , and S1 acts freely on Mc . Proposition 1. The hypersurfaces Mc ⊂ U are nondegenerate with respect to the metric g. Moreover, S1 acts isometrically on (Mc , g), and Mc is a S1 -principal ¯ . If k > 0 then the projection map bundle over U ¯ , g¯ ) πc : (Mc , g) −→ ( U is a semi-Riemannian submersion for c = 12 . (If k < 0 then πc is an anti-isometry on horizontal vectors for c = 12 ) We will need a lemma. Let h = g + iω denote the Hermitian product on U which is defined by g. We let ξ(u) = u denote the position vector field on U . Lemma 2. ¯ i) h(ξ, · ) = 2∂k ii) g(ξ, · ) = dk iii) g(ξ, ξ) = 2k Proof. In the complex coordinates we have ξ = h= Consequently,

#

∂2F Im ∂zi ∂zj !

"

% (zj ∂z∂ j + z¯j ∂∂z¯j ) and

dzi ⊗ d¯ zj .

! 2 " ∂ F Im zi d¯ zj ∂zi ∂zj ! " # ∂ 2 F¯ i # ∂2F = − zi d¯ zj − zi d¯ zj 2 ∂zi ∂zj ∂ z¯i ∂ z¯j !# " # ∂ 2 F¯ ∂F i d¯ zj − zi d¯ zj = − 2 ∂zj ∂ z¯i ∂ z¯j ¯ = 2∂k

h(ξ, · ) =

#

This proves i). Now ii) follows from i) by calculating ¯ + ∂k) = dk . g(ξ, ·) = Re h(ξ, ·) = (∂k

(5)

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Equation iii) is implied by ii), taking into account that the function k is R>0 -homogeneous of degree 2. Proof of Proposition 1. We consider the g-orthogonal decomposition Tu Cn+1 = Vu ⊕Hu into vertical and horizontal space which is defined by the canonical submersion ¯ . Then Vu is the real span of ξ and Jξ, and in fact Hu = {X ∈ Tu Cn+1 | π:U →U h(ξ, X) = 0}. In particular, g(ξ, X) = 0, for X ∈ Hu . Therefore, by ii) from the lemma, it follows that Hu ⊂ ker dk = T Mc . We compute the pull back π ∗ g¯ of the ¯ on the tangent space of Mc . Using (4) we get that special K¨ ahler metric g¯ on U ∗ gu (u, u) π g¯u = gu on Tu Mc . Now, by iii) of the lemma, gu (u, u) = 2k(u). The proposition follows. Proposition 3. The vector field ξ, which is the position vector field on the conic complex domain U , is also the position vector field in the affine coordinates xi , yi . Proof. To see this, we compute dxi (ξ) = Re dzi (ξ) = Re zi = xi " ! # ∂2F ∂F ∂F (ξ) = Re zj = Re = yi . dyi (ξ) = Re d ∂zi ∂zi ∂zj ∂zi % ∂ ∂ Hence, ξ = xi ∂x + yi ∂y as claimed. i i

Metric cones. For any manifold M with a (pseudo-) Riemannian metric g, the manifold C(M ) = R>0 × M with the metric dr2 + r2 g is called the metric cone over M . More generally, we consider cone metrics of the type gκ = κ1 dr2 +r2 g, where κ $= 0 is a constant. We denote the corresponding metric cone as Cκ (M ) = (C(M ), gκ ). Let us put sign k = 1 if k > 0 and sign k = −1 if k < 0. Corollary 4. Let U be a conic special K¨ ahler domain with K¨ ahler potential k, and special K¨ ahler metric g. Then (U, g) is isometric to the metric cone Csign k (M 12 ). Proof. Since R>0 acts freely on U , the map Φ : C(M 12 ) → U (r, u) +→ ru ∂ ) = ξ. The homogeneity of the holomorphic is a diffeomorphism. Note that dΦ(r ∂r potential F implies that the second derivatives of F are constant on radial lines in U . Hence, by formula (5), we have gru (rX, rX) = r2 gu (X, X), for u ∈ M 21 , X ∈ Tu M 12 . Moreover, by iii) of Lemma 2, gru (ξ, ξ) = 2k(ru) = r2 sign k. It is now immediate from ii) of Lemma 2 that Φ is an isometry.

Proposition 5. The one-form η := ω(ξ, ·) defines a contact structure on M 21 . Proof. dη = Lξ ω = 2ω is nondegenerate on ker η = Jξ ⊥ . 2.2. The affine geometry of conic special K¨ ahler domains. Having just seen that any conic special K¨ ahler domain (U, g) has the geometry of a metric cone over the level surface (M 21 , g) of k, we consider now the question how the flat affine connection ∇ on U interacts with the cone structure of (U, g). The flat affine geometry on U is determined by the coordinate change (2) which embeds U as a domain in R2n+2 . Since the symplectic form ω is ∇-parallel, so is the volume form θ = θg =

1 ω ∧n+1 . (n + 1)!


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Using the flat special coordinates we may view M 12 −→ R2n+2 immersed into affine space as a real hypersurface. In the light of Corollary 4, the next result shows that the metric structure on (U, g) is determined by the affine geometry of the hypersurface M 21 . Theorem 6. In the flat special coordinates of the special K¨ ahler domain U ⊂ Cn+1 the hypersurface M 12 ⊂ U immerses as a non-degenerate hypersurface in R2n+2 . The transversal field E = −sign k ξ is a Blaschke-normal for M 21 with respect to the volume form θ on R2n+2 , and the corresponding Blaschke-metric on M 12 coincides with the metric g induced from (U, g). Moreover, M 21 is an affine hypersphere of affine mean curvature sign k. We start the proof of the theorem with a lemma. Any vector field X on M 21 with ˜ on U which is defined by X(ru) ˜ values in Cn+1 has a natural extension X = rX(u), for u ∈ M 12 . Lemma 7. ˜ Y˜ ) = 2g(X, ˜ Y˜ ), i) ξ · g(X, ˜ Y˜ ) = 0, ii) (∇ξ g)(X, ˜ Y˜ ), if Y is tangent to M 1 . iii) g(ξ, ∇X˜ Y˜ ) = −g(X, 2 ˜ Y˜ ) is R>0 -homogeProof. Using Proposition 3, i) follows since the function g(X, ˜ ˜ ˜ = neous of degree 2. Also from X(ru) = rX(u), for all u ∈ U , we deduce that ∇ξ X ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ∇X˜ ξ = X. Therefore, (∇ξ g)(X, Y ) = ξ · g(X, Y ) − g(∇ξ X, Y ) − g(X, ∇ξ Y ) = ξ · ˜ Y˜ ) − 2g(X, ˜ Y˜ ). Hence, ii) follows from i). g(X, ˜ Y˜ ) = −(∇ ˜ g)(ξ, Y˜ ) = Now, if Y is tangent to M 21 then g(ξ, ∇X˜ Y˜ ) + g(X, X ˜ Y˜ ), by the symmetry of ∇g. Hence, iii) follows from ii). −(∇ξ g)(X,

Proof of Theorem 6. Let X, Y denote vector fields tangent to M 12 , and put κ = sign k. Then, by ii),iii) of Lemma 2, and Lemma 7 the Gauß-formula (1) for the hypersurface M 21 , with respect to ξ reads ˆ X Y − κg(X, Y ) ξ , ∇X Y = ∇

ˆ defines the induced connection on M 1 . Therefore the affine metric on M 1 where ∇ 2 2 with respect to the transversal vector field E = −κ ξ coincides with the metric g. Let θ 12 denote the metric volume form of the pseudo-Riemannian manifold (M 12 , g). To show that E is a Blaschke normal, we note that (for an appropriate choice of orientation of M 12 ) the metric volume form θ = θg of the ambient space (U, g) is given by θ = −κ dr ∧ r(2n+1) θ 21 in the conic product coordinates Φ from the proof of ∂ Corollary 4. And, therefore, θ(E, . . .) = θ(−κr ∂r , . . .) = θ 12 along M 21 . Hence, E is a 1 Blaschke-normal. Since A = −∇E = κ Id, M 2 is an affine hypersphere of affine mean curvature H = κ. Now it is easy to find a ∇-potential for g. Corollary 8. The K¨ ahler potential k is also a ∇-potential for the special K¨ ahler metric g, i.e. g = ∇dk on U . ˜ and Y˜ , we compute Proof. For homogeneous vector fields X ˜ · dk(Y˜ ) − dk(∇ ˜ Y˜ ) . (∇X˜ dk)(Y˜ ) = X X

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If Y is tangent to M 12 then, using ii) of Lemma 2 dk(∇X˜ Y˜ ) = g(ξ, ∇X˜ Y˜ ), and iii) ˜ Y˜ ). If Y˜ = ξ then we get (∇ ˜ dk)(ξ) = of Lemma 7 implies (∇X˜ dk)(Y˜ ) = g(X, X ˜ ˜ ˜ ˜ ˜ = g(ξ, X). ˜ X · dk(ξ) − dk(X) = X · 2k − dk(X) = dk(X) 3. Affine Sasakian hyperspheres. In Riemannian geometry a manifold (S, g) is called Sasakian if the corresponding metric cone (C(S), g1 ) is a K¨ ahler manifold, see e.g. [BG]. More generally, we call a (pseudo-) Riemannian manifold Sasakian if the metric cone Cκ (S) is a (pseudo-) K¨ ahler manifold. Let U be a a conic special K¨ ahler domain, and S = M 21 ⊂ U the affine sphere which is associated to U by Theorem 6. By Corollary 4, the affine hypersphere S is a Sasakian manifold. However, the concept of Sasakian manifold does not take into account the presence of the affine ˆ on S. Let (S, g, ∇) ˆ be a proper affine sphere. We show below that connection ∇ the metric cone Cκ (S) admits, as the natural affine differential geometric structure induced from S, the geometry of a parabolic affine hypersphere (C(S), gκ , ∇). This parabolic sphere is called the parabolic cone over S. In [BC] it was remarked that the geometric data of a special K¨ ahler manifold are in fact the geometric data of a parabolic sphere (M, ∇, g) whose Blaschke metric is K¨ ahler, and whose K¨ ahler form ω is ∇-parallel. This motivates the following

ˆ is called an affine Sasakian Definition 9. A proper affine hypersphere (S, g, ∇) hypersphere if the parabolic cone (C(S), g, ∇) over S is K¨ ahler, and the corresponding K¨ ahler form ω is ∇-parallel. Equivalently, a proper affine sphere S is affine Sasakian, if and only if the parabolic cone over S is special K¨ ahler. 3.1. The parabolic cone over a proper affine sphere. We show here that every proper affine hypersphere may be naturally realized as a hypersurface in a conic parabolic affine sphere. We already encountered this phenomenon, however in the particular context of conic special K¨ ahler domains. Proper spheres embed into conic parabolic spheres. Let (M, g) be a pseudoRiemannian manifold. We view M = {1} × M in a canonical way as a submanifold of Cκ (M ) with the metric g induced from the cone metric gκ on C(M ). Note also that the multiplicative group R>0 acts on C(M ). Proposition 10. Let ψ : S −→ Rn+1 be a proper affine hypersphere of affine ˆ h). Then the metric cone mean curvature κ, and with induced Blaschke data (∇, >0 Cκ (S) admits a torsionfree, flat, R -invariant connection ∇ so that the data (hκ , ∇) satisfy the integrability conditions for a parabolic affine hypersphere. Proof. We consider the local diffeomorphism Φ : C(S) → Rn+1 given by (r, u) +→ rψ(u) and let ∇ be the pullback of the canonical flat connection on Rn+1 . To simplify the notation we view S as a hypersurface in Rn+1 . Also we may then assume that E = −κ ξ is the affine normal of S, where ξ(x) = x is the position vector field on ¯ denote the constant extension of X to the Rn+1 . For a vector field X on S, let X ˜ on U = Φ(C(S)) product manifold C(S) = R>0 × S. Also we define the vector field X ∂ ˜ denote the position vector by X(ru) = rX(u), where u ∈ S and r > 0. We let ξ¯ = r ∂r ¯ = Φ∗ X, ˜ and ξ¯ = Φ∗ ξ. field on the cone C(S). Then X We show first that the metric volume form θhκ is ∇ parallel. Note first that, for the right choice of orientation of C(S), 1

θhκ = |κ|− 2 dr ∧ rn θh .


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We choose a (local) basis of vector fields X1 , . . . , Xn on S. Since θh (X1 , . . . , Xn ) = det(E, X1 , . . . , Xn ) along S, we get on C(S): 1

1

¯X ¯1, . . . , X ¯ n ) = |κ|− 2 rn+1 θh (X1 , . . . , Xn ) = ±|κ| 2 det(ξ, X ˜1, . . . , X ˜n) . θhκ (ξ, 1

Therefore θhκ = ±|κ| 2 Φ∗ det, and hence the equiaffine condition ii) is satisfied with respect to ∇. Next we show that ∇hκ is totally symmetric. It is enough to verify that (∇X hκ )(Y, Z) = (∇Y hκ )(X, Z) , for all vector fields X, Y and Z on C(S). We remark that if X, Y are vector fields on S the following formulas hold on C(S): ˆ X Y − κr−2 hκ (X, ¯ Y¯ )ξ¯ , ∇X¯ Y¯ = ∇ ¯ ¯ ¯ ¯ ∇X¯ ξ = X , ∇ξ¯X = X

(6) (7)

Therefore ¯ ¯ =X ¯ · hκ (Y¯ , Z) ¯ − hκ (∇X¯ Y¯ , Z) ¯ − hκ (Y¯ , ∇X¯ Z) (∇X¯ hκ )(Y¯ , Z) ˆ X Y, Z) − r2 h(Y, ∇ ˆ X Z) . = r2 X · h(Y, Z) − r2 h(∇ ¯ Y¯ , Z¯ the compatibility condition i) for hκ is implied by i) Hence, for vector fields X, for h. Next we compute ¯ ¯ = ξ¯ · hκ (Y¯ , Z) ¯ − hκ (∇ξ¯Y¯ , Z) ¯ − hκ (Y¯ , ∇ξ¯Z) (∇ξ¯hκ )(Y¯ , Z) ¯ − hκ (Y¯ , Z) ¯ − hκ (Y¯ , Z) ¯ =0. = 2hκ (Y¯ , Z) But also ¯ Z) ¯ ∇Y¯ Z) ¯ Z) ¯ = −hκ (∇Y¯ ξ, ¯ − hκ (ξ, ¯ (∇Y¯ hκ )(ξ, ¯ ξ)h ¯ κ (Y¯ , Z) ¯ + κr−2 hκ (ξ, ¯ =0. = −hκ (Y¯ , Z) ¯ = (∇X¯ hκ )(ξ, ¯ ξ) ¯ = 0. Hence, it follows that ¯ ξ) Finally, we easily see that (∇ξ¯hκ )(X, ∇hκ is totally symmetric. Note that (hκ , ∇) satisfies the integrability condition for parabolic spheres since ∇ is flat. Hence, (C(S), hκ , ∇) has the structure of a parabolic affine sphere. As a consequence of the fundamental theorem of affine differential geometry, if C(S) is simply connected, the data (hκ , ∇) are obtained from a Blaschke immersion Φ : ˆ is C(S) → Rn+2 as a parabolic affine hypersphere. Thus, the affine sphere (S, h, ∇) realized in a canonical way as a submanifold of a parabolic affine sphere (C(S), hκ , ∇), and the Blaschke metric on S, with respect to (C(S), ∇), coincides with the metric h, induced from hκ . We call the parabolic affine sphere (C(S), hκ , ∇) the parabolic cone over S. Completeness of affine spheres. We recall an important fact about parabolic spheres. Calabi [Ca1] and Pogorelov [Po] proved that if the affine metric g of a parabolic affine hypersphere (M, g, ∇) is definite and complete, then M must be a ˆ has a definite metric is also paraboloid. The case that a proper affine sphere (S, h, ∇) of particular interest. The Blaschke normal of S may be chosen so that the affine mean curvature H = κ is positive. If (with this choice of normal) the metric h is positive definite, then S is called an elliptic affine sphere, if h is negative definite then S

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is called hyperbolic. Therefore S is elliptic, if and only if the metric cone Cκ (S) carries a definite metric hκ . In the hyperbolic case the metric hκ has Lorentzian signature (1, n). There is the following result of Calabi [Ca2] on complete elliptic hyperspheres: Theorem 11. Let S be an elliptic affine hypersphere with complete Blaschke metric h. Then S is an ellipsoid. Let S be an elliptic affine hypersphere with complete metric h. Then the parabolic sphere (C(S), hκ , ∇) has definite metric hκ . However, clearly the metric cone C(S) is not complete. But Calabi’s theorem implies that if S is complete then C(S) = U ⊂ ¯ = Rn+1 , so that the metric hκ smoothly Rn+1 may be completed in 0 ∈ Rn+1 to U n+1 extends to R . We deduce: Corollary 12. Let S be an elliptic affine hypersphere with complete metric h and affine mean curvature κ. Then the parabolic cone (Cκ (S), hκ , ∇) is obtained by deleting a point in an elliptic paraboloid. 3.2. Characterization of affine Sasakian hyperspheres. Let (S, g) be a (pseudo-) Riemannian manifold, D the Levi-Civita connection on S. Then a Sasakian structure on S is provided by a Killing vector field σ of constant length g(σ, σ) = κ−1 so that the covariant derivative Φ = Dσ satisfies (DX Φ)(Y ) = κ(g(σ, Y )X − g(X, Y )σ) . The Killing vector field σ and the one-form η = κ g(σ, ·) are called the characteristic vector field and the characteristic one-form of the Sasakian structure on S. Let Cκ (S) ∂ be a metric cone over S, and let ξ = r ∂r denote the Euler field on C(S). We define a complex structure J on C(S) by the formulas ¯ = ΦX − η(X)ξ , Jξ = σ . JX

It is straightforward to verify that in fact J 2 = −Id, and that the cone metric gκ is J-invariant. Moreover J is parallel with respect to the Levi-Civita connection. Hence, J is integrable and Cκ (S) is K¨ ahler. Conversely, if Cκ (S) is K¨ ahler with respect to the complex structure J then σ = Jξ defines the characteristic vector field of a Sasakian structure on S. ˆ be a proper affine hypersphere with Sasakian Proposition 13. Let (S, g, ∇) ˆ is special K¨ structure σ. Then the parabolic cone over (S, g, ∇) ahler with respect to ˆ the complex structure J induced from σ if and only if Φ = ∇σ.

Proof. Let us first recall the formulas (6), (7) from the proof of Proposition 10, which are satisfied by the flat connection ∇ on C(S). Note also that the same ¯ where D ¯ is the (warped product) relations hold for the metric connections D and D, Levi-Civita connection of the cone metric gκ . Next we remark that the parabolic cone (C(S), gκ , ∇) is special K¨ ahler if and only if the special K¨ ahler condition d∇ J = 0

(8)

is satisfied. For a vector field Y on S, we compute (∇ξ J)Y¯ = 0, and (∇Y¯ J)ξ = ˆ Y σ − κg(Y, σ)ξ. Therefore if (8) ∇Y¯ σ ¯ − J Y¯ , where J Y¯ = DY σ − η(Y )ξ and ∇Y¯ σ ¯=∇ ˆ · σ = D· σ = Φ. Conversely, from Φ = ∇ ˆ · σ we is satisfied (∇Y¯ J)ξ = 0, and hence ∇ deduce that J Y¯ = ∇Y¯ σ ¯ and hence, since ∇ is flat, it follows (8) along S. Moreover, from the above equations d∇J(Y¯ , ξ) = (∇Y¯ J)ξ = 0 follows immediately. Therefore, the parabolic cone (C(S), gκ , ∇) is special K¨ ahler. ˆ we call σ an affine Consequently, if the Sasakian structure σ satisfies Φ = ∇σ ˆ Sasakian structure on the hypersphere (S, g, ∇).


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4. Applications. ¯ be a projective spe4.1. The Canonical circle bundle. Let π : M → M cial K¨ ahler manifold, where the conic manifold M carries the data (J, g, ∇). Let ˜ , J, g, ∇) be the universal covering space of M , and λ : M ˜ → V a compatible La(M grangian embedding into a pseudo-Hermitian, symplectic vector space (V, γ, Ω). Since the embedding λ is unique up to isometry of (V, γ, Ω), the function k(p) =

1 ˜ γ( λ(p), λ(p)) , p ∈ M 2

is invariant under deck-transformations of the covering, and hence defines a function k : M → R>0 . Note that, by iii) of Lemma 2 and by Corollary 8, (M, ∇, g) is a Hessian-manifold with potential k. We define a family of hypersurfaces Mc = {p ∈ M | k(p) = c} in M . Then the hypersurfaces Mc are invariant by the natural isometric S 1 ⊂ C∗ action on the conic manifold M . We call ¯ S := M 21 −→ M ¯. the canonical circle bundle over the projective special K¨ ahler manifold M ¯ be a projective special K¨ ¯ its Theorem 14. Let M ahler manifold and S → M canonical circle bundle. Then S has a canonical structure of a proper affine hypersphere. Moreover, S carries an affine Sasakian structure which determines the ¯. projective special K¨ ahler geometry on M Proof. It is enough to prove the theorem locally. Therefore we assume M 21 ⊂ U , where U is a special K¨ ahler domain with data (g, J, ∇). By Theorem 6, S = M 12 ⊂ U is a proper affine sphere, so that (U, g) is the metric cone over S. Since the flat coordinates on U are conic, i.e. R>0 -equivariant, the flat connection ∇ on U = C(S) coincides with the flat connection on C(S) which is constructed in Proposition 10. Hence, (U, g, ∇) is the parabolic cone over S, and the parabolic cone is special K¨ ahler. In particular, the sphere S is affine Sasakian, and, by Proposition 13, the Sasakian structure σ on S induced from J is affine Sasakian. ¯ 4.2. Projective special K¨ ahler domains with a definite metric. Let U be a projective special K¨ ahler domain with a definite metric g¯ and F the potential function of the corresponding special K¨ ahler domain U ⊂ Cn+1 \{0} which carries the special K¨ ahler metric g defined by formula (5). Note that by formula (4) the function ¯ , however the signature of the metric g on U is −F induces the same metric g¯ on U inverted. ¯ with a definite metric g¯ Definition 15. A projective special K¨ ahler domain U is called of elliptic type if the metric g on U is definite. ¯ is an elliptic projective special K¨ We remark that if U ahler domain, then by for¯ must be positive definite. Moreover the affine hypersphere mula (4) the metric g¯ on U ¯ by Theorem 6 has a definite metric, and S is an elS ⊂ U which is associated to U ¯ is a projective special K¨ liptic affine hypersphere. Conversely, if U ahler domain with a negative definite metric g¯, then the associated affine hypersphere S has an affine metric with Lorentzian signature.

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Characterization of complex projective space. In [L] it was proved that a special K¨ ahler manifold M with a (positive) definite complete metric is flat. In fact, it may also be deduced from this result that any complete special K¨ ahler domain U ⊂ Cn with n a definite metric is just C with a Hermitian inner product. In the case of projective special K¨ ahler domains there are many (homogeneous) examples with a definite and complete metric known, for instance, the examples given in the section 5. Among elliptic special K¨ ahler domains though, the projective space CPn is characterized by its completeness property: ¯ ⊂ CPn be a projective special K¨ Theorem 16. Let U ahler domain of elliptic type ¯ = CPn and g¯ is homothetic to the Fubini-Study with a complete metric g¯. Then U metric on CPn . Proof. We may choose F on U ⊂ Cn+1 so that g is positive definite. Therefore the K¨ ahler potential k on U is positive. By Theorem 6, the associated affine hypersphere ¯ is a Riemannian S is of elliptic type with a positive definite metric and, since S −→ U 1 submersion with a complete base and compact fibre S , S has a complete metric as well. Hence S is an ellipsoid by Thm 11. Recall that, by Corollary 4, the special ¯ is the parabolic cone over S and, by Corollary 12, K¨ ahler domain U ⊂ Cn+1 over U n+1 U =C \{0}. Also by Corollary 12, the metric g on U has a quadratic potential with respect to the flat connection ∇ on U . Since, by Corollary 8, k is a ∇-potential for g, k must be a homogeneous quadratic function in the affine coordinates. Hence, it follows that the cone metric g is parallel with respect to ∇. Therefore ∇ = D, which is possible only if F is a quadratic function and g is just a Hermitian inner product on Cn+1 . In this case, g¯ is homothetic to the Fubini-Study metric. 4.3. Calabi-Yau moduli space. We recall that a Calabi-Yau m-fold (of general type) is an oriented compact Riemannian manifold (X, g) with holonomy group Hol(X, g) = SU(m). This implies that X admits a unique complex structure J compatible with the orientation such that (X, J, g) is a K¨ ahler manifold and a parallel Jholomorphic (m, 0)-form vol (a holomorphic volume form), which is unique up to constant scale. In particular, (X, J) is a complex manifold of (complex) dimension m with ¯ be the Kuranishi moduli space of (X, J), trivial canonical bundle ∧m,0 T ∗ X. Let M i.e. the (local) moduli space of complex structures I on X. There is a natural holomor¯ whose fibre at I ∈ M ¯ is Γhol (∧m,0 T ∗ X) = H m,0 (X, I) (Γhol phic line bundle over M I ¯ be the corresponding holomorphic stands for holomorphic sections). Let π : M → M C∗ -bundle: π −1 (I) = H m,0 (X, I) − {0}. The√one-dimensional complex vector spaces & H m,0 (X, I) have a natural norm: -vol-2 := ( −1)−m X vol ∧ vol. Let S ⊂ M be the unit circle bundle with respect to that norm. ¯ be the above circle bundle over the Kuranishi moduli Theorem 17. Let S → M space of a Calabi-Yau threefold. Then S has naturally the structure of a Lorentzian affine Sasakian hypersphere. In particular, S is a proper affine hypersphere. ¯ has the structure of a projective special K¨ Proof. It is known that M ahler manifold. We briefly recall the construction of that structure. (For more details, see [C1]).√The cup product defines a complex symplectic form Ω on V := H 3 (X, C) and γ = −1Ω(·,¯·) is a pseudo-Hermitian form of (complex) signature (n + 1, n + 1), ¯ . The map where n = h1,2 = dim M ¯ / I +→ H 3,0 (X, I) ∈ P (V ) M is a holomorphic immersion and is induced by a conic holomorphic immersion φ : M → V − {0}, with the following properties: φ∗ Ω = 0 (φ is Lagrangian) and g = Re φ∗ γ


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is a K¨ ahler metric of complex signature (1, n) on the complex manifold M . These properties correspond to the first and second Hodge-Riemann bilinear relations for the underlying variation of Hodge structure of weight 3. As explained in section 1.2 the conic immersion φ induces on M the structure of a conic special K¨ ahler manifold ¯ is negative definite such that the corresponding projective special K¨ ahler metric on M (according to the conventions of this paper). Moreover, the circle bundle S defined above coincides with the canonical circle bundle S = M 21 of the projective special √ √ ¯ (notice that ( −1)−m = −1 for m = 3 and hence -u-2 = γ(u, u) K¨ ahler manifold M for u ∈ H 3,0 (X, I)). Now we can apply Theorem 14. 5. Homogeneous examples. The basic example of an affine Sasakian hypersphere S is provided by the total space of the Hopf fibration S = S 2n+1 = SU(n + 1)/SU(n) −→ CPn = SU(n + 1)/S(U(n)U(1)) . In the Lagrangian picture the corresponding conic affine special K¨ ahler manifold (M, J, g, ∇) is given as a linear Lagrangian subspace M ⊂ V = T ∗ Cn+1 for which the restriction of the Hermitian metric γ is positive definite. Since M is a linear subspace the flat connection ∇ coincides with the Levi-Civita connection D of g = Re γ. ¯ = CPn by holomorphic isometries of the The group SU(n + 1) acts transitively on M special K¨ ahler metric (Fubini-Study metric). The action is induced from the canonical linear symplectic action of SU(n + 1) on V = T ∗ Cn+1 which preserves the Hermitian metric γ and the Lagrangian subspace M ⊂ V . This action preserves also the affine Sasakian hypersphere S 2n+1 ⊂ M and induces a transitive action on S 2n+1 preserving the affine geometric and Sasakian structures. More generally, one can consider Lagrangian subspaces M ⊂ V = T ∗ Cn+1 of arbitrary Hermitian signature (p, q), p + q = n + 1. They correspond to fibrations ¯. S = SU(p, q)/SU(p, q − 1) −→ SU(p, q)/S(U(p, q − 1)U(1)) = M The case q = 1 is of particular interest. In that case the projective special K¨ ahler metric is negative definite (as for the Calabi-Yau moduli space and as for the target manifolds of N=2 D=4 supergravity theories with vector multiplets) and hence the ¯ = CHn is the metric of the affine Sasakian hypersphere has Lorentzian signature: M complex hyperbolic space and S is the real hyperbolic (2n+1)-space of Lorentzian signature (anti de Sitter space). ¯ = P (M ) will be called The Classification. A projective special K¨ ahler manifold M homogeneous if it admits a transitive group of isometries G whose action is induced by a G-action on the conic manifold M preserving the data (g, J, ∇). Homogeneous projective special K¨ ahler manifolds ¯ = P (M ) = G/K M with K compact have been classified in [AC] under the assumption that G is a real semisimple Lie group. We recall the result only in the most interesting case of nega¯ . It turns out that in this case the manifolds M ¯ = G/K are tive definite metric on M Hermitian symmetric spaces of non-compact type and are in one-to-one correspondence with the complex simple Lie algebras l different from cn = sp(C2n ). In all the cases the underlying conic affine special K¨ ahler manifold is a Lagrangian cone M ⊂ V generated by the G-orbit of a highest weight vector of a GC -module V of symplectic type. The GC -module V admits a G-invariant real structure τ compatible with the


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√ symplectic structure Ω, which defines a Hermitian metric γ = −1Ω(·, τ ·). The affine special K¨ ahler metric is the restriction of g = Re γ to M . The list is the following: ¯ = CHn = SU(n, 1)/S(U(n)U(1)), V = Cn+1 ⊕ (Cn+1 )∗ A) l = sln+3 (C), M ¯ = (SL(2, R)/SO(2)) × (SO(n − 1, 2)/SO(n − 1)SO(2)), BD) l = son+5 (C), M V = C2 ⊗ Cn+1 ¯ = SU(3, 3)/S(U(3)U(3)), V = '3 C6 E6) l = e6 (C), M E7) l = e7 (C),

¯ = SO∗ (12)/U(6), M

V (32) = V (π6 ) (semispinor)

¯ = E(−25) /E6 SO(2), M 7

V (56) = V (π1 ) ¯ = Sp(R6 )/U(3), V (14) (π3 ) = '3 C6 F) l = f4 (C), M 0 1 ¯ = CH = SL(2, R)/SO(2), V = (3 C2 . G) l = g2 (C), M

E8) l = e8 (C),

% Here V (λ) denotes the irreducible module GC -module with highest weight λ = λ i πi , where πi are the fundamental weights. The notation V (d) indicates that the module ¯ . The only has complex dimension d. Notice that in the cases A) and BD) n = dimC M redundancy in this list occurs the case n = 1. In fact, the Dynkin diagrams A3 = B3 define the same projective special K¨ ahler manifold CH1 = SU(1, 1)/S(U(1)U(1)) = SL(2, R)/SO(2). In both cases the corresponding conic manifold M is a linear Lagrangian subspace in the vector space V . Note that it may happen that projective special K¨ ahler manifolds are isometric as Riemannian manifolds, but nevertheless their special geometry is different: The ¯ = CH1 but in this case the underlying conic affine special diagram G2 defines M K¨ ahler manifolds M ⊂ V is not a linear subspace, as for type A), n=1. In fact, (3 2 V = C is the symmetric cube of the defining representation C2 of GC = SL(2, C). The Zariski closure of M ⊂ V is the nonlinear cone M ' = {u3 |u ∈ C2 } ⊂ V and M ⊂ M ' is open.

ˆ is called Homogeneous affine Sasakian spheres. An affine hypersphere (S, g, ∇) ˆ homogeneous if Aut(S) = Aut(S, g, ∇) acts transitively on S. Note that in general Aut(S) is a proper subgroup of Isom(S) = Aut(S, g). If S has an affine Sasakian structure σ then let Autσ (S) be the subgroup of those automorphisms in Aut(S) which commute with the flow of the vector field σ. We call Autσ (S) the group of automorphisms of the affine Sasakian sphere S. Clearly, any affine Sasakian hypersphere S with a transitive action of Autσ (S) is a circle bundle over a homogeneous projective ¯ is homogeneous then Isom(S) acts transitively on S. special K¨ ahler manifold. If M But note that, in general, the canonical isometric S 1 -action on S does not preserve ˆ The following theorem is a consequence of the above classification. the connection ∇. Theorem 18. Let S be the affine Sasakian hypersphere over a homogeneous ¯ = G/K of a real semisimple Lie group G. If projective special K¨ ahler manifold M ¯ is negative definite then M ¯ belongs to the above list the special K¨ ahler metric of M A)-G) and G acts transitively by automorphisms of the Lorentzian affine Sasakian hypersphere S. ¯ is induced by a G-action on the Proof. By construction, the G-action on M symplectic vector space V which preserves the geometric data on V . Hence G acts also on the canonical circle bundle S over M preserving the affine Sasakian geometry on S. Note now that in all the cases the centre Z(K) ∼ = U(1) of K acts non-trivially, ¯ over the canonical base point o = eK and hence transitively, on the fibre of S → M


129

¯ = G/K. This follows, for example, from the fact that K is the stabilizer of the in M line l = Cv ⊂ V generated by a highest weight vector v ∈ V τ of the GC -module V . In fact, K contains a (compact) Cartan subgroup of G, which cannot act trivially on l. (Notice, that the semisimple part of K, however, acts trivially on l.) Clearly, the Aut(S) action on the affine sphere S extends to a linear (with respect to the flat connection ∇) action on the parabolic cone which contains S. Hence, if G ⊂ Aut(S) acts transitively, the affine sphere S arises as a generic G-orbit in a real vector space W . If G is semisimple then S must be the level-set of a homogeneous G-invariant polynomial on W . Theorem 19. Let S be an affine Sasakian hypersphere with Lorentzian metric. ˆ If Autσ (S) contains a semisimple transitive group G then the affine sphere (S, g, ∇) arises as a hypersurface which is defined by a G-invariant homogeneous quartic polynomial on a real vector space W . Proof. S identifies with the canonical circle bundle in the parabolic cone M = C(S) which is special K¨ ahler. The action of Autσ (S) on S extends to an action on M which preserves the special K¨ ahler data on M . Using a compatible Lagrangian immersion we may therefore as well assume that the action of G = Autσ (S) on S is induced by an action of G on a Hermitian symplectic vector space (V, γ, Ω). In fact, we identify S as an affine sphere in the real vector space W = V τ , and S is a level set of the K¨ ahler potential k, which is, as a function on V τ , homogeneous of degree 2 and invariant by G. We claim that k 2 is a quartic polynomial. Since G acts with cohomogeneity one, it is sufficient to show that V τ admits a homogeneous G-invariant quartic polynomial, which is then necessarily proportional to k 2 . To show this it is clearly enough to construct a (complex) homogeneous GC -invariant quartic polynomial on V . The existence of such a polynomial on V follows by the following general argument. As we know, the GC -module V is associated to a Dynkin diagramm ∆ of the type A, B, D, E, F or G. We give some more detail how this correspondence works. (See [AC] for a complete account.) Let N = N (∆) = L/Lo be the compact symmetric quaternionic K¨ ahler manifold which is associated to the Dynkin diagramm ∆. (See [Wo].) L is the compact simple Lie group with trivial centre associated to ∆ and Lo = Sp(1)H is the stabilizer of a point o ∈ N . The complexified isotropy representation is a product To N ⊗ C = C2 ⊗C V . The group Sp(1) acts by the standard representation on C2 , and V is a complex module for H which admits a skew symmetric bilinear invariant. It follows that the maximal semisimple subgroup H ' ⊂ H is a compact form of a complex semisimple group GC which acts on V . In this way, we have associated a GC -module V to the Dynkin diagramm ∆. Now the quaternionic Weyl tensor, see [Sa], of the quaternionic K¨ ahler manifold N = N (∆) at the point o ∈ N gives rise to a nonzero GC -invariant element Q ∈ S 4 V ∗ . This shows the existence of a nontrivial GC -invariant homogeneous quartic polynomial Q on V . In examples it is not difficult to guess the quartic invariant Q directly from the G-module V τ . This gives an explicit description of the corresponding affine hyperspheres. Examples. A) G = SU(n, 1), V τ = Cn,1 , Q(v) = g(v, v)2 , where g is the SU(n, 1)-invariant Hermitian product.


130

BD) G = SL(2, R) × SO(n − 1, 2), V τ = R2 ⊗ Rn−1,2 ∼ = Hom((R2 )∗ , Rn−1,2 ). Let 2 ω be a SL(2, R)-invariant symplectic form on R and g the SO(n − 1, 2)-invariant scalar product, defining identifications Φω : R2 ∼ = (Rn−1,2 )∗ . = (R2 )∗ , Φg : Rn−1,2 ∼ 2 ∗ n−1,2 ∗ n−1,2 ∗ 2 For A ∈ Hom((R ) , R ) let A ∈ Hom((R ) , R ) be the dual morphism. Then Q(A) = det(A∗ Φg AΦω ). E6) GC = SL(6, C), V = (∧3 C6 )∗ . To any 3-form α we can associate the operator 6

Aα : C −→

)

5 *

6

C

+∗

= C6 ,

v +→ α ∧ ιv α .

Then Q(α) = trace(A2α ). It is easy to check that Q $= 0 by evaluating Q on dz 1 ∧ dz 2 ∧ dz 3 + dz 4 ∧ dz 5 ∧ dz 6 . This example is discussed in detail in [H] and the corresponding real symplectic SL(6, R)-module is also considered. Here we are interested in the real structure τ invariant under the real form G = SU(3, 3) of SL(6, C). It is induced by the SU(3, 3)-invariant pseudo-Hermitian form on C6 = C3,3 . In fact, this form induces a G-invariant pseudo-Hermitian form γ on V = (∧3 C6 )∗ . This determines a G-invariant real structure τ on V such that −iγ(·, τ ·) = Ω is the GC -invariant symplectic form of V : Ω(α, β)dz 1 ∧dz 2 ∧dz 3 ∧dz 4 ∧dz 5 ∧dz 6 = α∧β. '3 '3 6 '5 6 F) G = Sp(R6 ), V τ = 0 R6 is the kernel of the map R / α +→ ω ∧ α ∈ R , where ω is the symplectic form on R6 . The G-invariant quartic polynomial Q is just '3 6 the restriction of the SL(6, C)-invariant quartic polynomial on C , see previous '3 6 example, to the subspace 0 R .

(3 2 G) G = SL(2, R), V τ = R . The elements of V τ can be considered as homogeneous (2 2 2 cubic polynomials p on R . Let q(p) = det(∂ 2 p) ∈ R be the determinant of the Hessian of p. Then Q = D(q(p)) is the discriminant of q(p). Remarks: 1) In all the above examples (A-E) the group R∗ · G acts with an open orbit on V τ , in other words V τ with the action of R∗ · G is a real prehomogeneous vector space. Complex irreducible prehomogeneous vector spaces were classified in [SK]. Another description of the quartic invariant for the complex prehomogeneous vector spaces associated to some of the complex simple Lie algebras was recently given in [Cl]. 2) The Sasaki field σ of the affine hypersphere S ⊂ V τ can be easily computed from the real quartic invariant Q. From Lemma 2 ii) it follows that σ = Jξ is precisely the Hamilton vector field Xk associated to the K¨ ahler potential k. We can normalize the G-invariant real symplectic structure on V τ (or the invariant Q) such that Q is related to the K¨ ahler potential k by the formula Q = k 2 . Then we have XQ = 2kXk and therefore, since k = 1/2 on S, we have σ = XQ on S. Compact quotients. Let G be one of the real semi-simple Lie groups from the list A)-G). By Theorem 18, G acts transitively and properly on a Lorentzian affine ˜ which fibers over a Hermitian symmetric space G/K of nonhypersphere S = G/K ˜ compact type, K = KZ(K). By a result of Borel [Bo], G admits cocompact lattices Γ ≤ G. This allows to construct compact Clifford-Klein forms ˜ SΓ = Γ \ G/K


131

˜ The spaces SΓ admit an isometric for the Lorentzian homogeneous spaces S = G/K. 1 S -action (induced from the affine Sasakian structure) with finite stabilizers, the orbit space being a Hermitian locally symmetric space ¯ Γ = \ G/K . M Γ In his influential paper [Kul], Kulkarni observed the existence of non-trivial circle bundles over compact locally complex hyperbolic spaces, carrying a Lorentzian metric of constant curvature 1. This corresponds to the complex hyperbolic case ¯ = CHn = SU(n, 1)/S(U(n)U(1)), i.e. case A) in our list. In this sense, our conM struction generalizes Kulkarni’s construction of compact Lorentzian space-forms. It seems worthwile to further study the particular Lorentzian geometry of the homogeneous spaces S occuring in examples B) to G), and their compact Clifford-Klein forms. However, in this paper we content ourselves with summarizing what was just explained: ¯ be one of the Hermitian symmetric spaces appearing in Corollary 20. Let M ¯ Γ a compact Clifford-Klein form for M ¯ . Then M ¯ Γ is the orbit the list A)-G), and M space of an isometric S 1 -action on a compact Clifford-Klein SΓ for the Lorentzian ¯. homogeneous space S associated to M Acknowledgements. We thank Philipp Lohrmann and the anonymous referee for helpful remarks. REFERENCES [AC]

[ACD] [BC] [BG] [Bo] [Ca1] [Ca2]

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